Intrinsic circle domains

TL;DR

This paper proves that any finite connectivity, finite genus Riemann surface can be conformally mapped to an intrinsic circle domain, with a generalization to infinite connectivity and methods for numerical approximation.

Contribution

It establishes the existence and uniqueness of intrinsic circle domains for Riemann surfaces and introduces numerical methods for their approximation.

Findings

Existence and uniqueness of intrinsic circle domains for finite connectivity surfaces

Generalization to countably infinite connectivity surfaces

Numerical approximation techniques using circle packings and conformal welding

Abstract

Using quasiconformal mappings, we prove that any Riemann surface of finite connectivity and finite genus is conformally equivalent to an intrinsic circle domain U in a compact Riemann surface S. This means that each connected component B of S \ U is either a point or a closed geometric disc with respect to the complete constant curvature conformal metric of the Riemann surface (U union B). Moreover the pair (U,S) is unique up to conformal isomorphisms. We give a generalization to countably infinite connectivity. Finally we show how one can compute numerical approximations to intrinsic circle domains using circle packings and conformal welding.